A Simple Criterion for the Non-Existence of Limit Cycles of a Liénard System

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7th International Scientific Conference on Applied Sciences and Engineering 27-28 February, 2016 Flora Grand Hotel, Dubai

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Abstract of Applied Sciences and Engineering, 2016, Vol.7
DOI: 10.18488/journal.1001/2016.7/1001.7
7th
International Scientific Conference on Applied
Sciences and Engineering
27-28 February, 2016
Flora Grand Hotel, Dubai
Conference Website: www.scihost.org
2
Paper ID: 06/16/ 7
th
ISCASE
A Simple Criterion for the Non-Existence of Limit Cycles of a
Liénard System
Makoto HAYASHI1
1
Dept. of Math., College of Science and Technology, Nihon University 7-24-1, Narashinodai,
Funabashi-shi, Chiba, 274-8501, Japan
Abstract
In this paper, as an application in our result, the non-existence of limit cycles for the
Liénard system )(),( xgyxFyx   with










12)(5
1)(
)(
22
2
xforeexx
xforexx
xF
x
x
, )(xg x
is discussed by the simple criterion.
J.Graef[Gr :1971] has studied the uniformly boundedness of the solution orbits under
the condition
(C1) )()()(  xxGxF , 
x
dgxG
0
.)()( 
Moreover, he also proved the existence of limit cycles under the conditions (C1) and
(C2) .t. ( ) ( ) ( ) ( )
Recently, M.Cioni and G.Villari[2015] gave the same result as in [Gr] under the
conditions (C1) and
(C3) s.t. ( ) ( ) ( ) ( )
Note that (C2) is included to (C3). Our aim here is to discuss on the case of which (C1)
is satisfied, but (C3) is not satisfied. We shall give the simple criterion for the non-
existence of limit cycles for a Liénard system with these conditions.
Keywords: Liénard system, Limit cycle, Global asymptotic stability

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A Simple Criterion for the Non-Existence of Limit Cycles of a Liénard System